The idea of holographic superconductor was proposed by Hartnoll,
et.al., [1]} through building a holographic \(s\) -wave
superconductor, in the background of 4-dimensional Schwarzschild
anti-de Sitter (AdS) black holes. The motivation was to shed light
on the problem of high temperature superconductors. The most
well-known theory of superconductors was proposed by Bardeen,
Cooper and Schrieffer (BCS), which can successfully explain the
mechanism of the low temperature superconductors. According to BCS
theory, the condensation of pairs of electrons with antiparallel
spins (Cooper pairs) interacting through the exchange of phonon,
into a boson-like state [2]}. For building the
holographic superconductor, the correspondence between AdS spaces
and Conformal Field Theory (CFT) plays a crucial role
[1]}, [4]}. According to the AdS/CFT dictionary, a strong
coupling \(d\) -dimensional conformal field theory living on the
boundary is equivalent to the weak coupling gravity theory in \((d+ 1)\) -dimensional AdS bulk and each quantity in the bulk has a
dual on the boundary [5]}, [6]}, [7]}, [8]}, [9]}. In order
to describe a superconductor at boundary in holographic scenario,
we need a hairy black hole in the bulk. More precisely, we need a
hairy black hole (superconducting phase) for temperatures bellow
the critical value and a black hole with no hair (normal
phase/conductor phase) for upper values. During this process, the
system undergoes the spontaneous \(U(1)\)  symmetry breaking. The
condensation of a charged operator at the boundary corresponds to
emerge of the hair for black hole in the bulk. The quantum
description of the hair for black hole is the gas of charged
particles which have the same sign charge as the black hole. These
are repelled away by black hole and forbidden to escape to
infinity due to the presence of the negative cosmological constant
in the AdS bulk [10]}. The holographic superconductor theory
grabs a lot of attentions in the past decade (see e.g.
[11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}). Moreover, holographic superconductors
have also been explored widely in the regime of nonlinear
electrodynamics (see e.g.
[30]}, [16]}, [15]}, [18]}, [24]}, [25]}, [31]}, [32]}, [42]}, [43]}).
There are several types of nonlinear electrodynamics such as BI
[44]}, Exponential [45]}, Logarithmic [46]} and
Power-Maxwell [16]}, among them the most famous one is the
BI nonlinear electrodynamics which was first proposed for solving
the divergency in the electrical field of the point particles
[44]}, [49]}, [50]}, [51]}, [52]}. The studies on the holographic
superconductors have also generalized to other types such as
\(p\) -wave superconductors. The \(p\) -wave superconductivity is a
phase of matter where produces when the electrons are bounded with
parallel spins by exchange of the electronic excitations with
angular momentum \(\ell = 1\)  and condense in a triplet state. The
terms odd parity superconductivity, \(p\) -wave superconductivity and
triplet superconductivity all are equivalent [53]}.
Various models of holographic \(p\) -wave superconductors have been
investigated. Holographic \(p\) -wave superconductors can be studied
by condensation of a charge vector field in the bulk which
corresponds to the vector order parameter in the boundary [54]}, [55]}. This implies that the spin-1 order parameter can
be corresponded to the condensation of a 2-form field in the
gravity side[56]}. In [57]}, this type of
holographic superconductors characterized by introducing a \(SU(2)\) 
Yang-Mills gauge field in the bulk which one of the gauge degrees
of freedom corresponds to the vector order parameter at the
boundary. In addition, an alternative method to describe this kind
of holographic superconductor emerges by adopting a complex vector
field charged under a \(U(1)\)  gauge field which is equivalent to a
strongly coupled system involving a charged vector operator with a
global \(U(1)\)  symmetry at the boundary [58]}. For
this type of holographic superconductor, by decreasing temperature
below the critical value, the normal phase becomes unstable and we
observe the formation of vector hair which corresponds to
superconducting phase. Other investigations on the holographic
\(p\) -wave superconductors have been carried out in
(e.g.[59]}, [60]}, [61]}, [62]}, [63]}, [64]}, [58]}).
